IODS course project
RStudio Exercise 1: Tools and methods
The name of the course is Introduction to Open Data Science and we are focusing to language R, RStudio, GitHub and Markdown. You can find my Github repository here.
RStudio Exercise 2: Analysis
Introduction to the data
After the data wrangling exercise the new data set is found from the data folder. The set is based on data that was collected from course Introduction to Social Statistics, fall 2014 - in Finnish. The survey was conducted 3.12.2014 - 10.1.2015 by Kimmo Vehkalahti.
student2014 <- read.table("data/learning2014.txt", header = TRUE)
dim(student2014)## [1] 166 7The student data includes 7 variables and 166 rows.
str(student2014)## 'data.frame': 166 obs. of 7 variables:
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
## $ age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ attitude: int 37 31 25 35 37 38 35 29 38 21 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ points : int 25 12 24 10 22 21 21 31 24 26 ...Variables deep, strat and surf are combination variables from the original survey data.
summary(student2014)## gender age attitude deep stra
## F:110 Min. :17.00 Min. :14.00 Min. :1.583 Min. :1.250
## M: 56 1st Qu.:21.00 1st Qu.:26.00 1st Qu.:3.333 1st Qu.:2.625
## Median :22.00 Median :32.00 Median :3.667 Median :3.188
## Mean :25.51 Mean :31.43 Mean :3.680 Mean :3.121
## 3rd Qu.:27.00 3rd Qu.:37.00 3rd Qu.:4.083 3rd Qu.:3.625
## Max. :55.00 Max. :50.00 Max. :4.917 Max. :5.000
## surf points
## Min. :1.583 Min. : 7.00
## 1st Qu.:2.417 1st Qu.:19.00
## Median :2.833 Median :23.00
## Mean :2.787 Mean :22.72
## 3rd Qu.:3.167 3rd Qu.:27.75
## Max. :4.333 Max. :33.00From 166 survey respondents 56 was men and 110 was females. The mean of age was 25,5 years. Oldes respondet was 55 and youngest 17 years old.
Graphical output
Variables differ between genders. Distributions are different in age, attitude and surf (surfface). Three highest correlation between variables are:
- points-attitude
- surf-deep
- surf-attitude
Explanatory variables
Three variables
##
## Call:
## lm(formula = points ~ attitude + stra + surf, data = student2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.1550 -3.4346 0.5156 3.6401 10.8952
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.01711 3.68375 2.991 0.00322 **
## attitude 0.33952 0.05741 5.913 1.93e-08 ***
## stra 0.85313 0.54159 1.575 0.11716
## surf -0.58607 0.80138 -0.731 0.46563
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.296 on 162 degrees of freedom
## Multiple R-squared: 0.2074, Adjusted R-squared: 0.1927
## F-statistic: 14.13 on 3 and 162 DF, p-value: 3.156e-08In this linear regression model points are the target variable and attitude, strategy (stra) and surfface (surf) are explanatory variables.
Residuals of the model are between ~ -17.2 and ~10.9 when median is 0.52. I assume that errors are normally distributed but distribution needs confirmation.
Attitude is the only variable that has a very good significance in this model. p-value of stra and surf is too high to be even slightly significance.
Variables estimated coefficient is ~0.34 and it’s standard error is clearly smaller (~0.057). Other explanatory variables have not significance in this model.
Residual standard error is high in relation to a first and third quantiles of residuals.
This linear regression model of three explanatory variables explains ~19.3% of the points.
One variable
##
## Call:
## lm(formula = points ~ attitude, data = student2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.9763 -3.2119 0.4339 4.1534 10.6645
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.63715 1.83035 6.358 1.95e-09 ***
## attitude 0.35255 0.05674 6.214 4.12e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared: 0.1906, Adjusted R-squared: 0.1856
## F-statistic: 38.61 on 1 and 164 DF, p-value: 4.119e-09Estimated coefficient of explanatory variable attitude is ~0.35. This means that when attitude rises one point target variable (points) grows 0.35 times.
Attitude p-value (0.00000000412) shows that variable is very significant in this linear regression model
This model explains 18.6% of the points (target variable)
Graphical model validation
Residuals vs Fitted This plot shows that there is no pattern between residuals. There is a constant variance among errors. One can confirm that the assumption of constant variance of errors is valid.
Normal Q_Q Normal QQ-plot shows that the errors of the model are normally distributed.
Residual vs Leverage Last plot of the graphical model validation shows that the impact of the singel observation is moderate. Model includes some outliers but the leverage of singel observation don’t compromise the validation of the model.
RStudio exercise 3: Logistic regression
Introduction to the data
Using Data Mining To Predict Secondary School Student Alcohol Consumption. Fabio Pagnotta, Hossain Mohammad Amran Department of Computer Science,University of Camerino
https://archive.ics.uci.edu/ml/datasets/STUDENT+ALCOHOL+CONSUMPTION
Data is rolled into one from Math course and Portuguese language course datasets. After the data wrangling exercise the new data set is found from the data folder.
alc <- read.table("data/student_alc.txt", header = TRUE)
dim(alc)## [1] 382 35The student data includes 35 variables and 382 rows.
- The variables not used for joining the two data have been combined by averaging (including the grade variables)
- ‘alc_use’ is the average of ‘Dalc’ and ‘Walc’
- ‘high_use’ is TRUE if ‘alc_use’ is higher than 2 and FALSE otherwise
High and low alcohol consumption and other variables
In this analysis I’m going to study the relationship of high/low alcohol consumption between sex and the following variables:
| Variable | Type | Description |
|---|---|---|
| age | numeric | student’s age |
| studytime | numeric, scale 1-4 | weekly study time |
| freentime | numeric, scale 1-5 | free time after school |
| absence | numeric | number of school absences |
First these relationships are observed from tables and graphics. Hypothesis are as follows:
Age
The age of high consumption of alcohol may differ between sex. The development of charcter differs between young people and this may affect on habbits of alcohol consumption.
H0: Age don’t affect on alcohol consumption
H1: There is difference in between level of alcohol consumption and age
#a jitterplot of high_use, sex and age
g1 <- ggplot(alc, aes(x = high_use, y = age, col = sex))
g1 + geom_jitter() + ggtitle("Age by alcohol consumption and sex")
It seems that there is randomnes of sex and age in both alcohol consumption groups.
#a boxplot of high_use, sex and age
g1 <- ggplot(alc, aes(x = high_use, y = age, col = sex))
g1 + geom_boxplot() + ggtitle("Age by alcohol consumption and sex") + xlab("High consumption group") + ylab("Age of student")
Means however show that young male students have lower mean of age in low consumption group and females in high consumption group.
Study time
High alcohol consumption may be related to time spend in studies because one can’t do both at the same time at least not successfully.
H0: Alcohol consumption do not affect on weekly study time
H1: Alcohol consumption affects on weekly study time (numeric: 1 - <2 hours, 2 - 2 to 5 hours, 3 - 5 to 10 hours, or 4 - >10 hours)
#barplot about study time
ggplot(alc, aes(studytime, fill = high_use)) + geom_bar(position = "fill") +
ggtitle("Barplot about study time grouped by high_use")+ xlab("Study time") + ylab("Propotions of students") + scale_y_continuous(name = waiver(), breaks = waiver(), minor_breaks = waiver(), labels = waiver(), limits = NULL, expand = waiver(), na.value = NA_real_, trans = "identity")
It seems that there is less high users in those students groups who spend more time in studying (3-4) than in those who spend less time in studying (1-2).
Free time
High alcohol consumption may be related to free time so that studets who have more free time are consuming more alcohol that studets who haven’t as much free time.
H0: Amount of free time do not affect on alcohol consumption among students
H1: Amount of free time affects on alcohol consumption among students
#barplot about free time
ggplot(alc, aes(freetime, fill = high_use)) + geom_bar(position = "fill") +
ggtitle("Barplot about free time grouped by high_use")+ xlab("Free time") + ylab("Propotions of students") + scale_y_continuous(name = waiver(), breaks = waiver(), minor_breaks = waiver(), labels = waiver(), limits = NULL, expand = waiver(), na.value = NA_real_, trans = "identity")
It seems that there are more students that are consuming alcohol high amounts in those students groups that have more free time than in those who have not as much free time.
Absences
High consumption of alcohol may cause absences.
H0: High consumption of alcohol do not affect on absences
H1: High consumption of alcohol does affect on absences
#a boxplot of high_use and absences
g1 <- ggplot(alc, aes(x = high_use, y = absences, fill = high_use))
g1 + geom_boxplot() + ggtitle("Absences by alcohol consumption") + xlab("High consumption group") + ylab("Absences")
It seems that the differences between alcohol consumption groups in absences are small. Mean of absences is higher in high consumption group but it may not be significant.
Logistic regression model
#the model with glm()
m <- glm(high_use ~ age + studytime + freetime + absences, data = alc, family = "binomial")
summary(m)##
## Call:
## glm(formula = high_use ~ age + studytime + freetime + absences,
## family = "binomial", data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.9649 -0.8267 -0.6238 1.0990 2.2861
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.39240 1.79306 -2.450 0.014299 *
## age 0.18543 0.10313 1.798 0.072183 .
## studytime -0.51842 0.15867 -3.267 0.001086 **
## freetime 0.33015 0.12430 2.656 0.007907 **
## absences 0.07856 0.02269 3.463 0.000535 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 465.68 on 381 degrees of freedom
## Residual deviance: 423.18 on 377 degrees of freedom
## AIC: 433.18
##
## Number of Fisher Scoring iterations: 4From the fitted model one can see that all explanatory variables except age are statistically significant with p-value < 0,01. Variable absence is also statistically sisgnificant with p-value < 0,001. It seems that age doesn’t explain whether or not a student is a high user of alcohol.
Odds ratio and confidence intervals
# compute odds ratios (OR)
OR <- coef(m) %>% exp
# compute confidence intervals (CI)
CI <- confint(m) %>% exp## Waiting for profiling to be done...# print out the odds ratios with their confidence intervals
cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.01237102 0.0003481843 0.3998333
## age 1.20373817 0.9852330457 1.4775972
## studytime 0.59545916 0.4321414645 0.8061535
## freetime 1.39118359 1.0938389338 1.7826010
## absences 1.08172440 1.0368164005 1.1336048Odds ratio (OR) and the 95% confidence interval (CI) shows that those students who have a low study time are almost two times as likely to be a high user of alcohol than those studets who have higher study time. Students that have more freetime are also more likely to be a high user of alcohol. Also absences are positively correlated with high use of alcohol. Confidence interval shows that age is not statistically significant (because the interval contains 1) and other variables are.
Predictive power of the model
Predictive power of the final logistic regression model is calculated without the statistically insignificant variable age.
#the model with glm() and without the age variable
m_final <- glm(high_use ~ studytime + freetime + absences, data = alc, family = "binomial")
summary(m_final)##
## Call:
## glm(formula = high_use ~ studytime + freetime + absences, family = "binomial",
## data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.9420 -0.8332 -0.6450 1.1266 2.1537
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.34105 0.56337 -2.380 0.017293 *
## studytime -0.50496 0.15691 -3.218 0.001290 **
## freetime 0.32626 0.12379 2.636 0.008401 **
## absences 0.08324 0.02262 3.680 0.000233 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 465.68 on 381 degrees of freedom
## Residual deviance: 426.47 on 378 degrees of freedom
## AIC: 434.47
##
## Number of Fisher Scoring iterations: 4#predict and add the answer and the prediction to the data (alc)
probabilities <- predict(m_final, type = "response")
alc <- mutate(alc, probability = probabilities)
alc <- mutate(alc, prediction = probabilities > 0.5)
#tabulate the target variable versus the prediction
table("High use" = alc$high_use, "Prediction" = alc$prediction)## Prediction
## High use FALSE TRUE
## FALSE 254 14
## TRUE 93 21Table shows that the model predict 254 true negatives, 21 true positives, 14 false negatives and 93 false postives. This is sometimes called “confusion table”
table("High use" = alc$high_use, "Prediction" = alc$prediction) %>% prop.table() %>% addmargins()## Prediction
## High use FALSE TRUE Sum
## FALSE 0.66492147 0.03664921 0.70157068
## TRUE 0.24345550 0.05497382 0.29842932
## Sum 0.90837696 0.09162304 1.00000000Propabilities of the same table shows that 90,8% is predicted to be false but only 66,5% of them is correct. 9,2% is predicted to be true but 5,5% of them realy are students with high use of alcohol.
#a plot of 'high_use' versus 'probability' in 'alc'
g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))
g + geom_point()
Average number of wrong predictions
#defining a loss function (mean prediction error)
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
#calling loss_func to compute the average number of wrong predictions in the (training) data
loss_func(class = alc$high_use, prob = alc$probability)## [1] 0.2801047The average number of wrong predictions in trainig data is 28%.
Cross validation
#computing the average number of wrong predictions in the (training) data
#loss_func(class = alc$high_use, prob = alc$probability)
#K-fold cross-validation
library(boot)
cv <- cv.glm(data = alc, cost = loss_func, glmfit = m_final, K = 10)
#average number of wrong predictions in the cross validation
cv$delta[1]## [1] 0.282722510-fold cross-validation gives good estimate of the actual predictive power of the model. Low value = good.
RStudio exercise 4: Clustering and classification
Introduction to the data
In this exercise we use Boston data from MASS-library. This dataset contains information collected by the U.S Census Service concerning housing in the area of Boston Mass. Data includes 14 variables and 506 rows.
## [1] 506 14## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...| variable | description |
|---|---|
| crim | per capita crime rate by town |
| zn | proportion of residential land zoned for lots over 25,000 sq.ft. |
| indus | proportion of non-retail business acres per town |
| chas | Charles River dummy variable (= 1 if tract bounds river; 0 otherwise) |
| nox | nitrogen oxides concentration (parts per 10 million) |
| rm | average number of rooms per dwelling |
| age | proportion of owner-occupied units built prior to 1940 |
| dis | weighted mean of distances to five Boston employment centres |
| rad | index of accessibility to radial highways |
| tax | full-value property-tax rate per $10,000 |
| ptratio | pupil-teacher ratio by town |
| black | 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town |
| lstat | lower status of the population (percent) |
| medv | median value of owner-occupied homes in $1000 |
Graphical overview of the data
Plot matrix of the data
There are some very intresting distributions fo variables in the plot matrix. Variable rad has high and low values so the plot shows that the values are consenrated either side of the plot. VAriable *
Plotted correlation matrix
Plotted correlation matrix shows that there is some high correlation between variables:
Correlation is quite clear between industrial areas (indus) and nitrogen oxides (nox). Industry adds pollution in the area. Industry seems to correlate also with variablrs like age, dis, ras and tax.
Nitrogen oxides (nox) correlations are very similar with industry (indus)
Crime rate (crim) seems to correlate with good accessibilitty to radial highways (rad) and value property (tax).
Old houses (age) and employment centers have also something common
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00Scaled data
All the variables are numerical so we can use scale()-function to scale whole data set.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865## [1] "matrix"Scaling the data makes variables look as if they are in the same range. Variables like black and tax were before scaling hundred fold compared to some other variables.
Creating a new categorical variable crime
Variable crim is the base of the new categorical variable crime.
| categories | quantile points |
|---|---|
| low | 0%-25% |
| med_low | 25%-50% |
| med_high | 50%-75% |
| high | 75%-100% |
Quantile points of the variable crim
## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610## crime
## low med_low med_high high
## 127 126 126 127## zn indus chas nox
## Min. :-0.48724 Min. :-1.5563 Min. :-0.2723 Min. :-1.4644
## 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723 1st Qu.:-0.9121
## Median :-0.48724 Median :-0.2109 Median :-0.2723 Median :-0.1441
## Mean : 0.00000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723 3rd Qu.: 0.5981
## Max. : 3.80047 Max. : 2.4202 Max. : 3.6648 Max. : 2.7296
## rm age dis rad
## Min. :-3.8764 Min. :-2.3331 Min. :-1.2658 Min. :-0.9819
## 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049 1st Qu.:-0.6373
## Median :-0.1084 Median : 0.3171 Median :-0.2790 Median :-0.5225
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617 3rd Qu.: 1.6596
## Max. : 3.5515 Max. : 1.1164 Max. : 3.9566 Max. : 1.6596
## tax ptratio black lstat
## Min. :-1.3127 Min. :-2.7047 Min. :-3.9033 Min. :-1.5296
## 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049 1st Qu.:-0.7986
## Median :-0.4642 Median : 0.2746 Median : 0.3808 Median :-0.1811
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332 3rd Qu.: 0.6024
## Max. : 1.7964 Max. : 1.6372 Max. : 0.4406 Max. : 3.5453
## medv crime
## Min. :-1.9063 low :127
## 1st Qu.:-0.5989 med_low :126
## Median :-0.1449 med_high:126
## Mean : 0.0000 high :127
## 3rd Qu.: 0.2683
## Max. : 2.9865Train and test sets
Training set contains 80% of the data. 20% is in the test set.
## [1] 389 185 205 136 385 249 150 262 384 119 95 478 495 157 280 8 469
## [18] 191 475 398 139 36 476 47 256 86 179 75 405 367 62 257 193 501
## [35] 426 13 90 39 146 407 499 438 122 296 17 187 470 274 152 369 473
## [52] 400 123 177 474 110 453 313 154 463 153 57 73 236 425 423 278 432
## [69] 305 240 314 87 190 21 344 133 323 115 16 121 445 409 441 67 144
## [86] 20 415 76 44 408 419 472 489 135 208 422 108 243 32 285 447 358
## [103] 372 365 25 189 293 69 93 209 265 373 272 371 304 68 271 128 46
## [120] 413 180 89 6 504 151 352 113 210 105 41 310 124 462 1 363 206
## [137] 33 234 357 496 142 461 303 167 225 403 289 264 51 241 195 58 54
## [154] 140 125 160 431 245 454 242 118 443 197 386 404 10 317 162 34 52
## [171] 173 500 362 505 349 397 14 421 315 479 259 418 440 22 132 325 399
## [188] 174 2 116 382 450 226 102 215 424 239 204 319 270 318 411 491 101
## [205] 261 321 378 273 222 277 337 302 216 416 335 26 396 244 484 457 480
## [222] 28 361 88 276 260 15 223 40 27 231 111 214 340 467 182 60 217
## [239] 350 279 203 56 92 166 43 486 131 5 228 485 168 220 19 30 127
## [256] 267 80 99 345 148 331 181 149 430 254 169 23 492 229 379 70 183
## [273] 138 341 348 300 172 198 380 194 390 246 268 346 356 330 83 490 100
## [290] 444 12 468 437 284 301 29 202 410 487 339 477 49 251 4 394 42
## [307] 292 460 428 391 9 175 287 395 94 85 435 31 370 163 97 176 117
## [324] 465 355 98 427 368 63 456 451 158 91 324 332 48 464 283 213 219
## [341] 401 498 351 308 207 252 402 374 269 298 334 212 255 59 159 106 377
## [358] 253 412 360 38 388 286 248 50 72 297 164 178 488 417 199 436 71
## [375] 420 320 439 78 184 147 442 343 156 64 347 359 342 353 295 387 104
## [392] 366 192 466 328 333 126 77 45 497 81 221 364 227Fitting the Linear Discriminant Analysis
First the linear discriminant analysis (LDA) is fitted to the train set. The new categorical variable crime is the target variable and all the other variables of the dataset are predictor variables.
After fitting we draw the LDA biplot with arrows.
## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2400990 0.2623762 0.2376238 0.2599010
##
## Group means:
## zn indus chas nox rm
## low 0.94892287 -0.9230350 -0.109974419 -0.8830429 0.43687127
## med_low -0.05219905 -0.2991465 -0.012331882 -0.5945417 -0.12440342
## med_high -0.38808681 0.1900097 0.137785540 0.4030179 0.09310575
## high -0.48724019 1.0170492 -0.009855719 1.0798759 -0.46200885
## age dis rad tax ptratio
## low -0.9124045 0.8916117 -0.7095543 -0.7431447 -0.4846995
## med_low -0.3729296 0.3895785 -0.5539047 -0.5069226 -0.0914510
## med_high 0.4264565 -0.3731486 -0.3920855 -0.2899811 -0.2416883
## high 0.8311714 -0.8755308 1.6388211 1.5145512 0.7815834
## black lstat medv
## low 0.37806001 -0.77452047 0.51699127
## med_low 0.34514358 -0.13410974 0.02361544
## med_high 0.05428073 0.04502413 0.18603070
## high -0.78893431 0.91433433 -0.67137558
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.12353308 0.64682644 -0.92214263
## indus -0.05284177 -0.21086983 0.51816379
## chas -0.06560184 -0.01418336 0.11214638
## nox 0.45794294 -0.85770339 -1.36068152
## rm -0.09372081 -0.11219485 -0.18602505
## age 0.30116870 -0.34629856 -0.08136403
## dis -0.05803413 -0.32768984 0.20821236
## rad 2.96790538 1.01608937 0.15592961
## tax -0.01265549 0.01995781 0.26969570
## ptratio 0.15686311 -0.09318475 -0.33204408
## black -0.14185620 0.03787123 0.18751896
## lstat 0.28121206 -0.30226494 0.31267417
## medv 0.24796989 -0.48816558 -0.19567366
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9482 0.0386 0.0132## [1] 4 2 1 3 4 2 3 3 4 2 1 4 3 3 2 2 4 2 4 4 2 1 4 2 1 1 1 1 4 4 2 1 2 2 4
## [36] 2 1 2 3 4 2 4 1 2 3 1 4 2 3 4 3 4 2 1 4 3 4 3 3 4 3 1 2 3 4 4 1 4 1 2
## [71] 3 1 2 3 1 3 3 2 3 1 4 4 4 1 4 3 4 2 2 4 4 4 2 3 2 4 2 2 3 1 4 4 4 3 3
## [106] 2 1 2 1 2 3 4 2 4 2 1 3 3 2 4 1 1 1 1 3 1 2 3 2 1 3 2 4 1 4 2 3 3 4 2
## [141] 3 4 2 3 3 4 1 3 2 2 1 1 1 3 2 3 4 2 4 2 2 4 1 4 4 2 3 3 3 1 2 2 4 2 1
## [176] 4 3 4 3 4 3 4 4 3 3 3 4 2 1 2 4 4 3 2 3 4 2 1 3 2 2 4 2 2 3 2 4 2 3 2
## [211] 1 1 2 4 1 3 4 2 3 4 4 3 4 1 2 3 3 3 1 3 3 2 2 1 4 1 2 1 1 1 1 1 1 3 2
## [246] 3 3 1 3 3 3 2 3 3 3 3 2 1 1 3 1 1 3 4 3 3 3 2 3 4 2 2 3 1 1 1 3 1 4 1
## [281] 4 2 3 1 2 1 1 2 1 4 2 4 4 1 1 3 1 4 4 1 4 2 2 1 4 2 1 4 4 4 2 2 1 4 1
## [316] 1 4 3 4 3 2 1 2 4 1 2 4 4 2 4 4 3 1 3 1 2 4 1 2 2 4 3 1 1 2 2 4 4 3 2
## [351] 1 3 1 2 3 2 4 2 4 4 1 4 1 2 2 2 1 3 1 4 4 1 4 2 4 3 4 2 2 3 4 1 3 2 1
## [386] 4 1 1 1 4 2 4 1 3 2 1 2 2 2 3 1 3 4 3
Predicting the classes
## predicted
## correct low med_low med_high high
## low 19 10 1 0
## med_low 0 13 7 0
## med_high 0 9 21 0
## high 0 0 0 22Prediction were quite good. There was some errors in the middle of the range but classes low and especially high were good. Only one correct representative of high class was predicted to med_low class.
K-means algorithm
I’m going to calculate what is the optimal number of clusters for Boston data. First I reload and scale the data. Variables need to be scaled to get comparable distances between observation.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865Next I calculate the distances between observations and determinen the number of clusters. 
One way to determine the number of clusters is to look how the total of within cluster sum of squares (WCSS) behaves when the number of clusters changes. WCSS was calculated from 1 to 15 clusters. The optimal number of clusters is when the total WCSS drops radically. It seems that in this case optimal number of clusters is two. However we are here to learn so I decided to analyse model with four clusters.
After determining the number of clusters I run the K-means alcorithm again. 
It seems that when the data is divided to four clusters there is some clear differences in distriputions of several variables. Crim, zn, indus and blacks are variables were one can distinguish clear patterns between clusters. Some variables (rad & tax) show that sometimes 1 or 2 clusters make a clear distripution but observation of other two clusters are ambigious and there is no clear pattern to be regognised.
BONUS: LDA using clusters as target classes
After loading the Boston dataset I scale it to get comparable distances.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv clust
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063 Min. :1.000
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989 1st Qu.:2.000
## Median : 0.3808 Median :-0.1811 Median :-0.1449 Median :3.000
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean :2.674
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683 3rd Qu.:3.000
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865 Max. :4.000Original Boston dataset is now scaled and the result of K-means clustering is saved to the variable clust
LDA with the clusters
Next the LDA is performed and the biplot with arrows is created
## Call:
## lda(clust ~ ., data = scaled_Boston)
##
## Prior probabilities of groups:
## 1 2 3 4
## 0.2114625 0.1304348 0.4308300 0.2272727
##
## Group means:
## crim zn indus chas nox rm
## 1 -0.3912182 1.2671159 -0.8754697 0.5739635 -0.7359091 0.9938426
## 2 1.4330759 -0.4872402 1.0689719 0.4435073 1.3439101 -0.7461469
## 3 -0.3894453 -0.2173896 -0.5212959 -0.2723291 -0.5203495 -0.1157814
## 4 0.2797949 -0.4872402 1.1892663 -0.2723291 0.8998296 -0.2770011
## age dis rad tax ptratio black
## 1 -0.6949417 0.7751031 -0.5965444 -0.6369476 -0.96586616 0.34190729
## 2 0.8575386 -0.9620552 1.2941816 1.2970210 0.42015742 -1.65562038
## 3 -0.3256000 0.3182404 -0.5741127 -0.6240070 0.02986213 0.34248644
## 4 0.7716696 -0.7723199 0.9006160 1.0311612 0.60093343 -0.01717546
## lstat medv
## 1 -0.8200275 1.11919598
## 2 1.1930953 -0.81904111
## 3 -0.2813666 -0.01314324
## 4 0.6116223 -0.54636549
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## crim 0.18113078 -0.5012256 -0.60535205
## zn 0.43297497 -1.0486194 0.67406151
## indus 1.37753200 0.3016928 1.07034034
## chas -0.04307937 -0.7598229 -0.22448239
## nox 1.04674638 -0.3861005 -0.33268952
## rm -0.14912869 -0.1510367 0.67942589
## age -0.09897424 0.0523110 0.26285587
## dis 0.13139210 -0.1593367 -0.03487882
## rad 0.65824136 0.5189795 0.48145070
## tax 0.28903561 -0.5773959 0.10350513
## ptratio 0.22236843 0.1668597 -0.09181715
## black -0.42730704 0.5843973 0.89869354
## lstat 0.24320629 -0.6197780 -0.01119242
## medv 0.21961575 -0.9485829 -0.17065360
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.7596 0.1768 0.0636
Biplot shows that variables indus, zn and medv are the most influencial separators for the clusters.
Super-bonus
3D plot where observations color is the crime classes of the train set
3D plot where observations color is based on the K-means clusters.
Colors of the both plots is based to four classes. It seems that K-means plot shows the different clusters more clearly than the plot that is based on the crime classification.